Zero-Inflated Poisson (ZIP) Models

When a player's zeros come from two fundamentally different sources—"no opportunity" and "had opportunity but produced zero"—standard Poisson fails. Zero-Inflated Poisson (ZIP) is a mixture model designed specifically for this reality.

The Two Types of Zeros

Consider a pocket quarterback's rushing yards:

• Structural Zero (No Opportunity): The game plan never called for scrambles. He stayed in the pocket all game and handed off or threw on every play.
• Count Zero (Had Opportunity): He scrambled twice but was tackled for losses or no gain.

These zeros look identical in the box score, but they come from completely different processes. ZIP models both.

The ZIP Model Structure

Parameters:

• π (pi): Probability of the "structural zero" state (no opportunity)
• 1 - π: Probability of entering the "count process"
• λ (lambda): Average of the Poisson count process (when opportunity exists)

ZIP Probabilities:

For positive integers k ≥ 1:

Interpretation:

• With probability π, the player is in the "no-opportunity" state → outcome is automatically 0
• With probability (1-π), the player enters the count process → outcome follows Poisson(λ)

The ZIP Expected Value

The expected value under ZIP combines both states:

This is crucial for matching your mean projection while allowing the model to properly distribute probability around zero.

Case Study: Stafford Rushing Yards (ZIP Approach)

Let's apply ZIP to Matthew Stafford's rushing yards prop:

Given:

• Projected mean: E[Y] = 1.41 yards
• Observed structural zero rate: π ≈ 0.647 (65% of games with ≤0 yards)

Step 1: Calculate λ from the mean constraint

Since E[Y] = (1 - π) × λ, we can solve:

λ = E[Y] / (1 - π)
λ = 1.41 / (1 - 0.647)
λ = 1.41 / 0.353
λ ≈ 4.0

What this means in plain English:

• Stafford is positive only about 35% of the time (1 - π = 0.353)
• But when he IS positive, the average positive result is ~4 yards (λ = 4.0)
• This combination gives us the overall mean of 1.41 yards

Step 2: Calculate P(Y ≥ 1)

P(Y = 0) = π + (1 - π) × e^(-λ)
P(Y = 0) = 0.647 + 0.353 × e^(-4.0)
P(Y = 0) = 0.647 + 0.353 × 0.018
P(Y = 0) = 0.647 + 0.006
P(Y = 0) ≈ 0.653 (65.3%)

Therefore:

P(Y ≥ 1) = 1 - P(Y = 0) = 1 - 0.653 = 0.347 (34.7%)

Step 3: Calculate Expected Value

At +134 odds (risk $100 to win $134):

EV = P(Over) × profit - P(Under) × stake
EV = 0.347 × 134 - 0.653 × 100
EV = 46.50 - 65.30
EV = -$18.80

Comparing ZIP to Naïve Poisson

Let's see why the naïve Poisson approach fails:

Naïve Poisson (λ = 1.41):

P(Y = 0) = e^(-1.41) ≈ 0.244 (24.4%)
P(Y ≥ 1) = 1 - 0.244 = 0.756 (75.6%)
EV = 0.756 × 134 - 0.244 × 100 = +$76.90

The difference: Poisson predicts 24% zeros; reality shows 65% zeros. That 41-percentage-point gap completely flips the bet.

When to Use ZIP

Use ZIP when zeros arise from two distinct sources:

1. "No opportunity" games (role, game plan, injury limitations)
2. "Had opportunity but got zero" games (normal count variance)

Best Applications for ZIP

Estimating ZIP Parameters from Data

Estimating π (Structural Zero Probability)

The simplest approach: use the observed frequency of zeros (or non-positive outcomes):

π ≈ (Games with outcome ≤ 0) / (Total games)

Estimating λ (Conditional Mean)

Once you have π, solve for λ to match your projected mean:

λ = E[Y] / (1 - π)

Or calculate directly from positive games:

λ ≈ Average(outcomes | outcome > 0)

Excel Implementation

Cell C1: π (structural zero probability)
Cell D1: λ (Poisson mean for count process)
Cell A1: k (outcome to calculate probability for)

P(Y = 0):     =C1+(1-C1)*EXP(-D1)
P(Y ≥ 1):     =1-(C1+(1-C1)*EXP(-D1))
              OR: =(1-C1)*(1-EXP(-D1))
P(Y = k):     =(1-C1)*POISSON.DIST(A1,D1,FALSE)
E[Y]:         =(1-C1)*D1